Pulse compression is a signal processing technique used in the field of radar to increase the distance resolution of measurement as well as the signal-to-noise ratio. The general idea is to obtain a long pulse, so as to preserve sufficient energy on reception, without however sacrificing resolution relative to a short pulse of equivalent power.
Its principle is as follows: a signal is generated, whose temporal support is relatively long so as to maximize the transmitted energy. However, this signal is modulated in such a way that, after matched filtering, the inter-correlation between the signal received and the various frequencies of the transmitted signal makes it possible to resolve the return signals from several targets which might overlap inside the distance that the length of the pulse represents. As each part of the pulse has its own frequency, the returns arising from each target are completely separate.
In order to utilize the various channels of multi-channel radars, for example for monopulse processing, it is necessary to balance the gain and the phase of each of the reception channels. For narrowband radars for which it may be assumed that the response of the reception pathway is uniform in the instantaneous band, the measurement of differential gain and differential phase can be performed simply on the basis of a continuous wave (CW) reference signal reinjected on each transmission frequency at which the radar is required to work.
Within the framework of high-resolution radars, the gain can no longer be assumed to be constant over the instantaneous band of the radar and the compensation principle used in narrowband is no longer appropriate. Moreover, wideband radars generally use pulse compression waveforms, and the gain variation, in terms of amplitude and phase, inside the instantaneous band induces a degradation of the sidelobes at the output of the pulse compression processing intended to optimize the compromise between the probability of detection and the false alarm rate.
In the known solutions, it is generally sought to perform a linear processing by finite impulse response (FIR) filtering, with the aim of maximizing the probability of detection and of minimizing the false alarm rate. When the noise is white, it may be demonstrated that the achieving of these two optima amounts to maximizing the signal-to-noise ratio; in this case, the optimum, so-called matched, filter is that whose frequency response is the complex conjugate of the spectral density of the signal.
If the noise is not white, it will be considered that it is still sought to maximize the signal-to-noise ratio. It is then possible to reduce to the previous case by using a noise whitening filter beforehand, the matched filter then being the cascade of the two filters. However, the whitening filter is in general not achievable, since it is non-causal and of infinite duration. It is therefore only possible to have an approximation thereof.
The optimum (in terms of signal-to-noise) filter of finite dimension equal to that of the signal has response −1 s*(−t) where  is the restriction to the duration of the signal of the noise power correlation matrix. If the input noise in the reception pathway is white and if the duration of the signal were infinite then −1 would correspond to the inverse of the power response of the pathway, that is to say to the equalization in terms of power of the received signal. The duration being finite, −1 is merely the best finite approximation of this equalization (with respect to the signal-to-noise).
−1 is not measurable in practice on the basis of a measurement on noise. Moreover, the exact response of the transmitted signal is not known exactly either, since it comprises the defects introduced by the transmission pathway.
It is therefore sought to achieve an approximation of the operation described −1 s*(−t) on the basis of an on-signal measurement (assumed reasonably devoid of noise by way of a coherent average).
Moreover, it is known that, in the case of dispersive pulses with linear frequency modulation, the signal obtained after matched filtering exhibits natural overshoots whose level (13 dB for the closest) may mask other targets. To reduce the level of these lobes, it is customary to weight the temporal response of the matched filter, this presenting the counterpart to degrading the signal-to-noise ratio. There is therefore a compromise between level of sidelobes and degradation of the signal-to-noise ratio SNR.
Finally, within the framework of multi-channel radars which comprise several independent reception pathways, it is fundamental that the responses of these pathways be identical both in amplitude and in phase. When the instantaneous band of the signal is low, the undulations in the response of the reception pathways remain very limited in the band so that these responses may be considered to be constant. It is then sought to equalize the responses with the aid of a differential gain and a differential phase. This is no longer true when the instantaneous band increases, above all when it is sought to use compact and selective filtering technologies (surface wave filters, ceramic filters, etc.), and the equalization must then take the form of a filtering. If the objective is to restore a flat response for each pathway, and if it is considered that the noise factor of the pathways occurs almost wholly at the head of the pathway, then this filtering corresponds to the whitening of the noise on the one hand and to a differential gain and a differential phase on the other hand.
For multi-channel high-resolution radars of the instantaneous wideband type, a Wiener filtering could be used to compensate the variations of complex gain in the band. The corresponding filter is obtained by calibration and applies to the totality of the temporal support considered. It consists in calculating the spectrum of the calibration signal as it exits the pathway, by discrete Fourier transform, and in inverting it. Applied to the useful signal containing the whole set of individual echoes of the radar, this filter yields, for each echo, a correlation spike of unit duration. This filter is therefore ideal, but it is non-causal and exhibits a temporal support at least equal to the duration of the useful signal, in general much greater than the duration of an individual echo. It therefore cannot, in particular, be embodied as an FIR, this nevertheless being imperative when the duration of the useful signal is very significant.